How can I get "Plane.000" instead of "Plane"?
I don't recommend doing this, BUT here is a way to change blenders default naming behaviour.  The following needs to be "tightened up" for other names eg will make mistakes with names like "Planet", but will do as a simple example.  

Open a new blender file, run the script below, then add or duplicate planes.  The scene_update_pre handler runs alot.  Any added plane, cube or sphere will be renamed to start at 000.

import bpy
prim_names = ["Plane", "Cube", "Sphere"]

def name_prims(scene):
    for name in prim_names:
        prims = [o for o in bpy.data.objects 
                 if o.name.startswith(name)]
        newprim = scene.objects.get(name)
        if newprim:
            newprim.name = "%s.%03d" % (name, len(prims) - 1)

# make it a handler
bpy.app.handlers.scene_update_pre.append(name_prims)


Remove all scene update pre handlers with: (Note this will remove all handlers, this one and those added by addons, etc)

# remove all handlers
bpy.app.handlers.scene_update_pre.clear()


Otherwise remove it by name

my_handlers = [h for h in bpy.app.handlers.scene_update_pre
               if h.__name__.startswith("name_prims")]

for h in my_handlers:
    bpy.app.handlers.scene_update_pre.remove(h)